Black-Scholes Equation

A Black-Scholes Equation is a partial differential equation (PDE) that governs the price evolution of European-style options under the Black–Scholes–Merton model.

• Context:
  • It can (typically) be used to model the dynamics of the option's price as a function of both time and the underlying asset's price changes.
  • It can (typically) assume a frictionless market, where there are no transaction costs or taxes, and money can be borrowed and lent at a risk-free rate.
  • It can incorporate the effects of constant volatility and risk-neutral valuation in pricing the option.
  • It can (often) serve as the foundation for deriving the Black-Scholes Formula, which provides a closed-form solution for the prices of European call and put options.
  • It can (typically) enable the calculation of Greeks, which are measures of the sensitivity of the option's price to various parameters.
  • It can be solved using numerical methods for options that do not have analytical solutions, such as American-style options.
  • It can (often) be extended or modified to accommodate more complex option pricing models that account for dividends, stochastic volatility, or jumps in asset prices.
  • It can (typically) form the basis for the no-arbitrage argument in financial mathematics, ensuring that the option pricing model does not allow for arbitrage opportunities.
  • ...
• Example(s):
  • the pricing of a non-dividend-paying European call option.
  • the use of numerical methods to price complex derivatives based on the equation's framework.
  • ...
• See: Black–Scholes Formula, Mathematical Finance, Black–Scholes–Merton Model, Option (Finance), Derivative (Finance), Volatility (Finance), Frictionless Market, Hedge (Finance).

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References

2024

• (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/Black–Scholes_equation Retrieved:2024-2-27.
  • In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of derivatives under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives.

    Consider a stock paying no dividends. Now construct any derivative that has a fixed maturation time [math]\displaystyle{ T }[/math] in the future, and at maturation, it has payoff [math]\displaystyle{ K(S_T) }[/math] that depends on the values taken by the stock at that moment (such as European call or put options). Then the price of the derivative satisfies : [math]\displaystyle{ \begin{cases} \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0 \\ V(T, s) = K(s) \quad \forall s \end{cases} }[/math] where [math]\displaystyle{ V(t, S) }[/math] is the price of the option as a function of stock price S and time t, r is the risk-free interest rate, and [math]\displaystyle{ \sigma }[/math] is the volatility of the stock.

    The key financial insight behind the equation is that, under the model assumption of a frictionless market, one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently “eliminate risk". This hedge, in turn, implies that there is only one right price for the option, as returned by the Black–Scholes formula.

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